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Lebesgue Sequence Spaces

  • René Erlín Castillo
  • Humberto Rafeiro
Chapter
Part of the CMS Books in Mathematics book series (CMSBM)

Abstract

In this chapter, we will introduce the so-called Lebesgue sequence spaces, in the finite and also in the infinite dimensional case. We study some properties of the spaces, e.g., completeness, separability, duality, and embedding. We also examine the validity of Hölder, Minkowski, Hardy, and Hilbert inequality which are related to the aforementioned spaces. Although Lebesgue sequence spaces can be obtained from Lebesgue spaces using a discrete measure, we will not follow that approach and will prove the results in a direct manner. This will highlight some techniques that will be used in the subsequent chapters.

References

  1. [6]
    A. L. Cauchy. Cours d’Analyse de l’Ecole Royale Polytechnique: Analyse Algébrique. Debure, 1821.zbMATHGoogle Scholar
  2. [26]
    G. H. Hardy. Note on a theorem of Hilbert. Math. Z., 6:314–317, 1920.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [27]
    G. H. Hardy. Note on a theorem of Hilbert concerning series of positive terms. Proc. Lond. Math. Soc. (2), 23:xlv–xlvi, 1925.Google Scholar
  4. [30]
    G. H. Hardy, J. E. Littlewood, and G. Pólya. Inequalities. Cambridge, at the University Press, 1952. 2d ed.Google Scholar
  5. [32]
    O. Hölder. Ueber einen Mittelwertsatz. Gött. Nachr., 1889:38–47, 1889.zbMATHGoogle Scholar
  6. [48]
    L. Maligranda. Why Hölder’s inequality should be called Rogers’ inequality. Math. Inequal. Appl., 1(1):69–83, 1998.MathSciNetzbMATHGoogle Scholar
  7. [51]
    H. Minkowski. Geometrie der Zahlen. I. Reprint. New York: Chelsea Co., 256 p. (1953)., 1953.Google Scholar
  8. [58]
    F. Riesz. Les systèmes d’équations linéaires à une infinite d’inconnues. Paris: Gauthier-Villars, VI + 182 S. 8. (Collection Borel.) (1913)., 1913.Google Scholar
  9. [61]
    L.J. Rogers. An extension of a certain theorem in inequalities. Messenger of mathematics, XVII(10):145–150, 1888.Google Scholar
  10. [82]
    H. Weyl. Singuläre Integralgleichungen mit besonderer Berücksichtigung des Fourierschen Integraltheorems. Göttingen, 86 S (1908)., 1908.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • René Erlín Castillo
    • 1
  • Humberto Rafeiro
    • 2
  1. 1.Universidad Nacional de ColombiaBogotáColombia
  2. 2.Pontificia Universidad JaverianaBogotáColombia

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